E.1 Calculus
Differentiation as gradient and rate of change. The power rule for integer exponents. Finding tangent and normal lines, increasing/decreasing intervals, critical points, and using these in optimization. Anti-differentiation and definite integrals as area under a curve. The trapezoidal rule for numerical area approximation.
The derivative as gradient and rate of changeSign up
Gradient at a Point · Derivative as a Rate
Increasing and decreasing functions; sign of f′Sign up
Sign of f' and Monotonicity
Power rule differentiation (integer exponents)Sign up
d/dx(axⁿ) = anxⁿ⁻¹ · Sums and constants
Tangent and normal linesSign up
Tangent gradient = f′(a) · Normal gradient = −1/f′(a)
Anti-differentiation and definite integralsSign up
Reverse of differentiation · Definite integral as area
Critical points where the gradient is zeroSign up
Local max, local min, stationary points · Sign-change test
Optimization in contextSign up
Setting up an objective · Solving f′(x) = 0 in real-world problems
Trapezoidal rule for area approximationSign up
Trapezoidal rule formula · Over/under estimation depending on concavity